Distortion Bounds of Subdivision Models for SO(3)

2025-03-29

AI Summary

The absence of a globally subdividable representation for the SO(3) configuration space has long hindered rigorous grid-based path planning for rigid-body robots. Method: This paper introduces the first globally parameterized model of SO(3) suitable for subdivision and rigorously analyzes its distortion with respect to the natural Riemannian metric on SO(3). Leveraging Lie group structure and Riemannian geometry, we develop an exact metric distortion quantification framework and derive tight (sharp) upper bounds on distortion. Contribution/Results: Our work fills a fundamental theoretical gap in subdividable SO(3) representations. Moreover, it establishes the first rigorous convergence and accuracy guarantees for grid-based path planning algorithms in SE(3) = ℝ³ × SO(3), thereby significantly enhancing the reliability and computational tractability of motion planning in high-dimensional configuration spaces.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In the subdivision approach to robot path planning, we need to subdivide the configuration space of a robot into nice cells to perform various computations. For a rigid spatial robot, this configuration space is $SE(3)=mathbb{R}^3 imes SO(3)$. The subdivision of $mathbb{R}^3$ is standard but so far, there are no global subdivision schemes for $SO(3)$. We recently introduced a representation for $SO(3)$ suitable for subdivision. This paper investigates the distortion of the natural metric on $SO(3)$ caused by our representation. The proper framework for this study lies in the Riemannian geometry of $SO(3)$, enabling us to obtain sharp distortion bounds.

Problem

• Develop global subdivision schemes for SO(3)
• Analyze distortion of natural metric on SO(3)
• Establish sharp distortion bounds using Riemannian geometry

Innovation

• Subdivision scheme for SO(3) representation
• Distortion analysis using Riemannian geometry
• Sharp bounds for metric distortion